The (Siegel) Eisenstein series of weight \(k\geq4\) is defined by the formula
\[
 \psi_k(Z)=\sum_{(C,D)}\det(CZ+D)^{-k},
\]
where the summation extends over all inequivalent bottom rows \((C,D)\) of elements of \({\rm Sp}(4,\mathbb{Z})\) with respect to left multiplication by elements of \({\rm SL}(2,\mathbb{Z})\).
By a classical result of <span class="name">Jun-Ichi Igusa</span> (On Siegel modular forms of genus two. Amer. J. Math. 84 (1962), 175-200, <a href="http://www.ams.org/mathscinet-getitem?mr==0141643">MR0141643</a>), the ring <script type="math/tex">M_{2*}({\rm Sp}(4,\mathbb{Z}))</script> of Siegel modular forms of degree 2 with <b>even weights</b> with respect to the full modular group Sp(4,Z) is generated by the Eisenstein series
\[
 \psi_4,\;\psi_6,\;\psi_{10}\;\psi_{12}.
\]
Alternatively, \(\psi_{10}\) can be replaced by the cusp form
\[
 \chi_{10}=-43867\cdot 2^{-12}3^{-5}5^{-2}7^{-1}53^{-1}(\psi_4\psi_6-\psi_{10})
\]
and \(\psi_{12}\) can be replaced by the cusp form
\[
 \chi_{12}=131\cdot593\cdot2^{-13}3^{-7}5^{-3}7^{-2}337^{-1}(441\psi_4^3+250\psi_6^2-691\psi_{12}).
\]
Note that \(\chi_{10}\) is a Saito-Kurokawa lifting coming from the cuspform of weight \(10\) for \({\rm SL}(2,\mathbb{Z})\), and \(\chi_{12}\) is a Saito-Kurokawa lifting coming from the cuspform of weight \(12\).

Thus the even weight forms are:
$${\Bbb C}[\psi_4,\psi_6,\chi_{10},\chi_{12}],$$
with the cusp forms being the ideal generated by $\chi_{10}, \chi_{12}$.

The odd weight forms are:
$$\chi_{35}\,{\Bbb C}[\psi_4,\psi_6,\chi_{10},\chi_{12}],$$
where $\chi_{35}$ is a weight 35 cusp form.
For a formula for $\chi_{35}$, see [...]
